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The classical Morse-Thom-Smale construction associates to a Morse function g : M → R on a closed manifold a chain complex C∗(g) of free abelian groups, where the number of free generators of Cp(g) equals the number of critical points of g of index p for each p. The homology of this complex is isomorphic to the homology of the manifold , and the boundary operator in this complex is defined in a geometric way, using the algebraic number of trajectories of a gradient of g, joining critical points of g (see [5], [9], [13], [14], [15]). In the early 80s S.P.Novikov generalized this construction to the case of maps f : M → S ( see [6]). Here M is a closed connected manifold , f : M → S is a Morse map, non-homotopic to zero. The corresponding analog of Morse complex is a free chain complex C∗(f) over the ring Z((t)) of the formal power series with integer coefficients and finite negative part (that is Z((t)) = { ∑ ant n | an ∈ Z and ∃N : an = 0 for n < N}). The number of free generators of Cp(f) equals the number of critical points of f of index p, and the homology of C∗(f) equals to the completed homology of the corresponding cyclic covering of M . The boundary operator in this complex depends on the choice of a Riemannian metric on M or of a gradient-like vector field v for f . We prefer in our work the language of gradient-like vector fields (see §2 for precise definitions). To explicit the boundary operators, let M̄ P −→ M be the connected infinite cyclic covering for which f ◦P is homotopic to zero. Choose a lifting F : M̄ → R of f ◦ P and let t be the generator of the structure group (≈ Z) of P such that for every x ∈ M̄ we have F (xt) < F (x). For every critical point x of f choose a lifiting x̄ of x to M̄ . Choose orientations of stable manifolds of critical points. Then for every critical points

Rationality and exponential growth properties of the boundary operators in the Novikov Complex